Two different natural numbers are selected from the set $\ \allowbreak \{1, 2, 3, \ldots, 6\}$. What is the probability that the greatest common factor of these two numbers is one? Express your answer as a common fraction.
Solution: We consider all the two-element subsets of the six-element set $\{ 1, 2, 3, 4, 5, 6 \}$. There are ${6 \choose 2} = 15$ such subsets. And of these, only the subsets $\{ 2, 4 \}, \{2, 6 \}, \{3, 6 \}, \{ 4, 6 \}$ are not relatively prime. So the probability of the two-element subset's elements having greatest common factor one is $1 - \frac{4}{15} =\boxed{ \frac{11}{15}}$.